# About probability - Rubix cube and Monty Phyton problem. Thank you...?

Hello everyone...

I'm curious about this. One of my lecturer gave an added question about the rubix cube during our homework (just for an extra credit). Needless to say, i didn't get that credit (i'm not that smart with probability).

The question, more or less, was :

"Starting from a fully-formed rubix cube, how many possible combination that you could get ? Please explain the answer with and without rotating and mirroring"

Rotating means looking at the cube from different face (without altering the cube) and mirroring means looking at the cube' reflection in the mirror.

Also, could anybody give me an explanation about Monty Phyton problem ?

Monty Phyton problem is :

"Suppose there're 3 doors. One of the doors has a prize. You could pick one (says you pick no.1). X knows the right door and will show you one of the wrong doors (suppose 2 is wrong). You're given one last chance. Go with no.1 or switch to no.3 ?"

The answer is switching to no.3. Why ?

Best regards,

-septerra-

Relevance

A normal (3×3×3) Rubik's Cube can have (8! × 38−1) × (12! × 212−1)/2 = 43,252,003,274,489,856,000 different positions (permutations), or about 4.3 × 1019, forty-three quintillion (short scale) or forty-three trillion (long scale), but the puzzle is advertised as having only "billions" of positions, due to the general incomprehensibility of such a large number to laymen. Despite the vast number of positions, all Cubes can be solved in twenty-seven or fewer moves (see Optimal solutions for Rubik's Cube).

To put this into perspective, if every permutation of a Rubik's Cube was lined up end to end, it would stretch out approximately 261 light years. If they were laid side by side, it would cover the Earth approximately 256 times.

In fact, there are (8! × 38) × (12! × 212) = 519,024,039,293,878,272,000 (about 519 quintillion on the short scale) possible arrangements of the pieces that make up the Cube, but only one in twelve of these are actually reachable. This is because there is no sequence of moves that will swap a single pair or rotate a single corner or edge cube. Thus there are twelve possible sets of reachable configurations, sometimes called "universes" or "orbits", into which the Cube can be placed by dismantling and reassembling it.

Source(s): wikipedia
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• Cant do the rubik's cube

Monty Phyton :

you pick one so leaving one of the following possibilities

Right Wrong

Wrong Right

Wrong Wrong

now X shows you a wrong door

if you choose the other you will be right 2/3 times , the only time you will be wrong was on the 1/3 chance that you picked the right door in the first place.

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• Anonymous

The answer for the doors problem is, as you said, switch to no.3. The reason is that the probability of each being the prize is 1/3. When the wrong door is removed (X shows you 2 is wrong, in your example) the probability of your door being WRONG is stilll 2/3, and so switching gives you a 2/3 probability of winning.

Source(s): An A-level Maths class last year. We discussed it and found this somewhat counter-intuitive reason!
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• for monty python (?) question:

in the beginning there are 3 choices:

- you choose 1 and have 1/3 (33%) chance of being correct

- the friend (does not matter if they want you to win or not) chooses one door that is NOT correct.

- eliminating one wrong door means the other now has a 2/3 chance of being correct (since the two doors each had 1/3 and we have eliminated one of them). your choices probability did not change. It is still at 1/3.

- so always switch to door 3

another way to look at it is:

- door 2 is 1/3

- door 3 is 1/3

- so you have 1/3 chance and the other two doors have 2/3 chance

- so if the friend gets two choices they have 2/3 chance of being correct

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